User blog:B1mb0w/The R Function
'The R Function' THIS BLOG HAS BEEN REPLACED BY MY BLOG ON The Rex Function WHICH IS MUCH STRONGER. The R function generates very large numbers. It is based on my earlier work on the The S Function. It has a growth rate \(\approx f_{LVO}(n)\). 'What is the R Function' The R Function is actually two functions \(R()\) and \(r()\) which use this simple ruleset: \(R(n) = R(0,n) = n + 1\) \(R(a + 1, n) = R^n(a,n_*)\) \(R(r(0), n) = R(n,n)\) and other instances of \(n\) can be substituted with \(r(0)\) \(r(a + 1) = R^{r(a)}(r(a)_*,r(a))\) and \(r(1, 0) = r^{r(0)}(0)\) \(r(1, a + 1) = r^{r(1, a)}(r(1, a))\) \(r(b + 1, 0) = r^{r(b, 0)}(b, 0_*)\) \(r(1, 0, 0) = r^{r(1, 0)}(1_*, 0)\) and \(R(1, 0, n) = R(r(1, 0_{r(0)}),n)\) 'Some Identities' Some R Function identities are: \(R(R(R(a,b)),b) > R(R(a,b),R(a,b))\) because \(R(R(R(a,b)),b) = R^b(R(a,b),b_*) = R(R(a,b),R^{b-1}(R(a,b),b_*))\) and \(R^{b-1}(R(a,b),b_*) > R(R(a,b),b) > R(a,b)\) 'Notation Explained' I use notation that is not in general use, but I find helpful. They are the \(*\) and parameter subscript brackets. The \(*\) notation is used to explain nested functions. For example: \(M(a) = M(a)\) \(M^2(a) = M(M(a))\) then let \(M^2(a,b_*) = M(a,M(a,b))\) \(M^2(a_*,b) = M(M(a,b),b)\) Parameter subscript brackets are useful for functions with many parameters: \(M(a) = M(a)\) \(M(a,b) = M(a,b)\) then let \(M(a,0_{1}) = M(a,0)\) \(M(a,0_{3}) = M(a,0,0,0)\) \(M(a,b_{2}) = M(a,b_1,b_2)\) \(M(a,0_{2},b_{3},1) = M(a,0,0,b_1,b_2,b_3,1)\) 'Growth Rate of the R Function ... to \(\Gamma_0\)' The R Function behaves like the FGH function up to a point: \(R^h(g,n_*) = f_g^h(n)\) \(R(r(0),n) = f_{\omega}(n)\) \(R(R(1,r(0)),n) = f_{\omega.2}(n)\) \(R(R(2,r(0)),n) = f_{\omega.2^{\omega}}(n)\) \(R(R(3,r(0)),n) = f_{\varphi(1,0)}(n)\) \(R(R(r(0),r(0)),n) \approx f_{\varphi(\omega,0)}(n)\) \(R(R(R^2(r(0)),r(0)),n) = f_{\varphi(\omega,0)}(n)\) \(R(R(R(3,r(0)),r(0)),n) = R(R^2(3_*,r(0)),n) \approx f_{\varphi(\varphi(1,0),0)}(n) = f_{\varphi^2(1_*,0)}(n)\) \(R(R^{r(0)}(3_*,r(0)),n) \approx f_{\varphi^n(1_*,0)}(n) = f_{\varphi(1,0,0)}(n)\) \(R(r(1),n) = R(R^{r(0)}(r(0)_*,r(0)),n) > R(R^{r(0)}(3_*,r(0)),n) \approx f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\) 'Growth Rate ... to \(\varphi(1,0,0,0)\)' The R Function will eventually reach and surpass the small Veblen ordinal (svo): \(R(R(r(0),r(1)),n) \approx f_{\varphi(\omega,\varphi(1,0,0)+1)}(n)\) \(R(R(R(r(0),r(0)),r(1)),n) \approx f_{\varphi(\varphi(\omega,0),\varphi(1,0,0)+1)}(n)\) \(R(R(R^{r(0)}(3_*,r(0)),r(1)),n) \approx f_{\varphi(1,0,1)}(n)\) Let \(r(1) > \alpha = R^{r(0)}(3_*,r(0)) \approx \varphi(1,0,0)\) \(R(R(\alpha,r(1)),n) \approx f_{\varphi(1,0,1)}(n)\) \(R(R(1,R(\alpha,r(1))),n) \approx f_{\varphi(1,0,1).2}(n)\) \(R(R(3,R(\alpha,r(1))),n) \approx f_{\varphi(1,\varphi(1,0,1)+1)}(n)\) \(R(R(r(0),R(\alpha,r(1))),n) \approx f_{\varphi(\omega,\varphi(1,0,1)+1)}(n)\) \(R(R(\alpha,R(\alpha,r(1))),n) \approx f_{\varphi^n(1_*,\varphi(1,0,1)+1)}(n) = f_{\varphi(1,0,2)}(n)\) or \(R(R^2(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,2)}(n)\) \(R(R^3(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,3)}(n)\) \(R(R^{r(0)}(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,\omega)}(n)\) \(R(R^{R(1,r(0))}(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,\omega.2)}(n)\) \(R(R^{R(2,r(0))}(\alpha,r(1)_*),n) > f_{\varphi(1,0,\omega^2)}(n)\) \(R(R^{R(3,r(0))}(\alpha,r(1)_*),n) > f_{\varphi(1,0,\varphi(1,0))}(n)\) \(R(R^{\alpha}(\alpha,r(1)_*),n) > f_{\varphi(1,0,\varphi(1,0,0))}(n)\) or \(R(R^{r(1)}(\alpha,r(1)_*),n) = R(R(\alpha),r(1)),n) > f_{\varphi^2(1,0,0_*)}(n)\) \(R(R(R^2(\alpha),r(1)),n) > f_{\varphi^3(1,0,0_*)}(n)\) \(R(R(R^{r(0)}(\alpha),r(1)),n) > f_{\varphi^n(1,0,0_*)}(n) = f_{\varphi(1,1,0)}(n)\) or \(R(R(R(1,\alpha),r(1)),n) > f_{\varphi(1,1,0)}(n)\) Let \(\beta = R(1,\alpha)\) \(R(R(\beta,r(1)),n) > f_{\varphi(1,1,0)}(n)\) \(R(R(R(\beta,r(1)),R(\beta,r(1))),n) > f_{\varphi(1,1,1)}(n)\) Using the identity defined above \(R(R(R(R(\beta,r(1))),r(1)),n) > f_{\varphi(1,1,1)}(n)\) \(R(R^2(R(R(\beta,r(1))),r(1)_*),n) > f_{\varphi(1,1,2)}(n)\) \(R(R^{r(0)}(R(R(\beta,r(1))),r(1)_*),n) > f_{\varphi(1,1,\omega)}(n)\) \(R(R^{r(1)}(R(R(\beta,r(1))),r(1)_*),n) > f_{\varphi(1,1,\varphi(1,0,0))}(n)\) \(R(R(R^2(R(\beta,r(1))),r(1)),n) > f_{\varphi(1,1,\varphi(1,0,0))}(n)\) \(R(R(R^{r(0)}(R(\beta,r(1))),r(1)),n) \approx f_{\varphi(1,1,\varphi(1,0,0).\omega)}(n)\) \(R(R(R^{r(1)}(R(\beta,r(1))),r(1)),n) \approx f_{\varphi(1,1,\varphi(1,0,0)^2)}(n)\) \(R(R(R^{R(\beta,r(1))}(R(\beta,r(1))),r(1)),n) \approx f_{\varphi^2(1,1,0_*)}(n)\) \(R(R(R(1,R(\beta,r(1))),r(1)),n) \approx f_{\varphi^2(1,1,0_*)}(n)\) \(R(R(R^2(1,R(\beta,r(1))_*),r(1)),n) \approx f_{\varphi^3(1,1,0_*)}(n)\) \(R(R(R^{r(0)}(1,R(\beta,r(1))_*),r(1)),n) > f_{\varphi^n(1,1,0_*)}(n) = f_{\varphi(1,2,0)}(n)\) or \(R(R(R(2,R(\beta,r(1))),r(1)),n) > f_{\varphi(1,2,0)}(n)\) or \(R(R(R(2,R(R(1,\alpha),r(1))),r(1)),n) > f_{\varphi(1,2,0)}(n)\) \(R(R(R(R(2,\alpha),r(1)),r(1)),n) > f_{\varphi(1,2,0)}(n)\) \(R(R(R(r(1),r(1)),r(1)),n) > f_{\varphi(1,2,0)}(n)\) \(R(R^2(r(1)_*,r(1)),n) > f_{\varphi(1,2,0)}(n)\) \(R(R^3(r(1)_*,r(1)),n) > f_{\varphi(1,3,0)}(n)\) \(R(R^{r(0)}(r(1)_*,r(1)),n) > f_{\varphi(1,\omega,0)}(n)\) \(R(R^{r(1)}(r(1)_*,r(1)),n) > f_{\varphi(1,\varphi(1,0,0),0)}(n) = f_{\varphi^2(1,0_*,0)}(n)\) \(R(r(2),n) > f_{\varphi^2(1,0_*,0)}(n)\) then \(R(R(r(1),r(2)),n) > f_{\varphi(1,\varphi(1,0,0) + 1,0)}(n)\) \(R(R^{r(0)}(r(1)_*,r(2)),n) > f_{\varphi(1,\varphi(1,0,0) + \omega,0)}(n)\) \(R(R(r(1)),r(2)),n) = R(R^{r(2)}(r(1)_*,r(2)),n) > f_{\varphi(1,\varphi^2(1,0_*,0),0)}(n) = f_{\varphi^3(1,0_*,0)}(n)\) \(R^2(R(r(1)),r(2)_*),n) > f_{\varphi^4(1,0_*,0)}(n)\) \(R^{r(0)}(R(r(1)),r(2)_*),n) > f_{\varphi^n(1,0_*,0)}(n) = f_{\varphi(2,0,0)}(n)\) or \(R(R^2(r(1)),r(2)),n) > f_{\varphi(2,0,0)}(n)\) Let \(\gamma = R^2(r(1))\) \(R(\gamma,r(2)),n) > f_{\varphi(2,0,0)}(n)\) \(R(R(R(\gamma,r(2)),R(\gamma,r(2))),n) > f_{\varphi(2,0,1)}(n)\) Using the identity defined above \(R(R(R(R(\gamma,r(2))),r(2)),n) > f_{\varphi(2,0,1)}(n)\) or \(R(R(R(R(\gamma),r(2)),r(2)),n) > f_{\varphi(2,0,1)}(n)\) \(R(R^2(R(\gamma)_*,r(2)),n) > f_{\varphi(2,0,1)}(n)\) \(R(R^2(R^3(r(1))_*,r(2)),n) > f_{\varphi(2,0,1)}(n)\) or \(R(R^2(r(2)_*,r(2)),n) > f_{\varphi(2,0,1)}(n)\) WORK IN PROGRESS 'Growth Rate ... to svo' The R Function will eventually reach and surpass the small Veblen ordinal (svo): WORK IN PROGRESS The following is earlier work from my S Function. \(S(n,g(1,0_{g(0)}),1) > S(n,g(1,0_{n-1}),1) \approx f_{\varphi(1,0_{n})}(n) = f_{svo}(n)\) 'Growth Rate ... to LVO' The Generalised S Function is one of the Fastest Computable functions: \(g(0) \approx \omega = \vartheta(0)\) \(S(g(0),3,1) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\) \(g(1) \approx \Gamma_0 = \varphi(1,0,0) = \vartheta(\Omega^2)\) \(g(1;0) > g(1,0_{g(0)}) \approx svo = \vartheta(\Omega^\omega)\) \(g(1;0_{g(0)}) \approx \vartheta(\Omega^\omega\omega)\) TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\) \(g_1(0) > g(1;0_{g(1;0)}) \approx \vartheta(\Omega^{\omega+1})\) \(g_1(1) \approx \vartheta(\Omega^{\omega+2})\) \(g_1^2(0) > g_1(g_1(0)) \approx \vartheta(\Omega^{\omega.2})\) \(g_1(1,0) \approx \vartheta(\Omega^{\omega.3})\) \(g_1(1;0) \approx \vartheta(\Omega^{\omega^2})\) \(g_1(1;0_{2}) = g_1(1;0;0) \approx \vartheta(\Omega^{\omega^3})\) \(g_1(1;0_{g(0)}) \approx \vartheta(\Omega^{\omega^{\omega}})\) \(g_2(0) \approx \vartheta(\Omega^{\omega^{\omega^{\omega}}}) = \vartheta(\Omega^{\omega\uparrow\uparrow 3})\) \(g_3(0) \approx \vartheta(\Omega^{\omega\uparrow\uparrow 4})\) \(g_{g(0)}(0) \approx \vartheta(\Omega^{\omega\uparrow\uparrow\omega}) = \vartheta(\Omega^{\varphi(1,0)})\) \(g_{S(g(0),1,1)}(0) \approx \vartheta(\Omega^{\varphi(1,1)})\) \(g_{S(g(0),2,1)}(0) \approx \vartheta(\Omega^{\varphi(1,\omega^2)})\) \(g_{S(g(0),3,1)}(0) \approx \vartheta(\Omega^{\varphi(1,\varphi(1,0))}) = \vartheta(\Omega^{\varphi^2(1,0_*)})\) \(g_{S(g(0),g(0),1)}(0) \approx \vartheta(\Omega^{\varphi(2,0)})\) \(g_{g(1)}(0) \approx \vartheta(\Omega^{\varphi(1,0,0)})\) \(g_{g(1;0)}(0) \approx \vartheta(\Omega^{\Omega})\) Large Veblen ordinal \(LVO ≥ f_{\vartheta(\Omega^\Omega)}(n)\) \(g_{g(2;0)}(0) \approx \vartheta(\Omega^{\Omega^2})\) Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n) = f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\) 'Further References' Further references to relevant blogs can be found here: User:B1mb0w Category:Blog posts